question_answer

\[\cos  \left( \alpha  - \beta  \right) = 1 and cos \left( \alpha  + \beta  \right) =1/e\], where \[\alpha ,\text{ }\beta \text{ }\in (-\pi ,\,\,\pi )\]. Pairs of a, P which satisfy both the equations is/are

A) 0                     

B)        1

C) 2                     

D)        4

Correct Answer: D

Solution :

Given that, \[\cos  (\alpha  - \beta ) = 1\] and \[\cos  (\alpha  - \beta ) = 1/e\], where a, \[\alpha ,\,\beta \,\,\in \,\,\left[ -\pi ,\,\,\pi  \right]\] Now, \[\cos \left( \alpha -\beta  \right)= 1\,\,\Rightarrow \,\,\,\alpha -\beta =0\,\,\Rightarrow \,\,\,\alpha =\,\,\beta \] \[\therefore \,\,\,cos(\alpha +\beta )=\,\,1/e\,\,\Rightarrow \,\,cos\,\,2\alpha =1/e\] \[\therefore  \,0 <1/e <1 and 2\alpha  \in  \left[ -2\pi ,\,\,2\pi  \right]\] There will be two values of \[2\alpha \] satisfying \[\cos  2\alpha  =1/e\,\,in \left[ 0,\,\,2\pi  \right] and two in \left[ -\,2\pi ,\,\,0 \right].\] There will be four values of a in \[[-\,\pi ,\,\,\pi ]\] and correspondingly four values of \[\beta \]. Hence there are four sets of \[\left( \alpha ,\, \beta  \right).\]